Calculus: Concepts and Connections

0 Preliminaries 0.1 Polynomial and Rational Functions 0.2 Graphing Calculators and Computer Algebra Systems 0.3 Inverse Functions 0.4 Trigonometric and Inverse Trigonometric Functions 0.5 Exponential and Logarithmic Functions 0.6 Transformations of Functions 0.7 Parametric Equations and Polar Coordinates 1 Limits and Continuity 1.1 A Brief Preview of Calculus 1.2 The Concept of Limit 1.3 Computation of Limits 1.4 Continuity and its Consequences 1.5 Limits Involving Infinity 1.6 Limits and Loss-of-Significance Errors 2 Differentiation 2.1 Tangent Lines and Velocity 2.2 The Derivative 2.3 Computation of Derivatives: The Power Rule 2.4 The Product and Quotient Rules 2.5 The Chain Rule 2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions 2.7 Derivatives of Exponential and Logarithmic Functions 2.8 Implicit Differentiation 2.9 The Mean Value Theorem 3 Applications of Differentiation 3.1 Linear Approximations and Newton's Method 3.2 Indeterminate Forms and L'Hopital's Rule 3.3 Maximum and Minimum Values 3.4 Increasing and Decreasing Functions 3.5 Concavity and Overview of Curve Sketching 3.6 Optimization 3.7 Rates of Change in Economics and the Sciences 3.8 Related Rates and Parametric Equations 4 Integration 4.1 Area Under a Curve 4.2 The Definite Integral 4.3 Antiderivatives 4.4 The Fundamental Theorem of Calculus 4.5 Integration by Substitution 4.6 Integration by Parts 4.7 Other Techniques of Integration 4.8 Integration Tables and Computer Algebra Systems 4.9 Numerical Integration 4.10 Improper Integrals 5 Applications of the Definite Integral 5.1 Area Between Curves 5.2 Volume 5.3 Arc Length and Surface Area 5.4 Projectile Motion 5.5 Applications of Integration to Physics and Engineering 5.6 Probability 6 Differential Equations 6.1 Growth and Decay Problems 6.2 Separable Differential Equations 6.3 Euler's Method 6.4 Second Order Equations with Constant Coefficients 6.5 Nonhomogeneous Equations: Undetermined Coefficients 7 Infinite Series 7.1 Sequences of Real Numbers 7.2 Infinite Series 7.3 The Integral Test and Comparison Tests 7.4 Alternating Series 7.5 Absolute Convergence and the Ratio Test 7.6 Power Series 7.7 Taylor Series 7.8 Applications of Taylor Series 7.9 Fourier Series 7.10 Power Series Solutions of Differential Equations 8 Vectors and the Geometry of Space 8.1 Vectors in the Plane 8.2 Vectors in Space 8.3 The Dot Product 8.4 The Cross Product 8.5 Lines and Planes in Space 8.6 Surfaces in Space 9 Vector-Valued Functions 9.1 Vector-Valued Functions 9.2 Parametric Surfaces 9.3 The Calculus of Vector-Valued Functions 9.4 Motion in Space 9.5 Curvature 9.6 Tangent and Normal Vectors 10 Functions of Several Variables and Differentiation 10.1 Functions of Several Variables 10.2 Limits and Continuity 10.3 Partial Derivatives 10.4 Tangent Planes and Linear Approximations 10.5 The Chain Rule 10.6 The Gradient and Directional Derivatives 10.7 Extrema of Functions of Several Variables 10.8 Constrained Optimization and Lagrange Multipliers 11 Multiple Integrals 11.1 Double Integrals 11.2 Area, Volume and Center of Mass 11.3 Double Integrals in Polar Coordinates 11.4 Surface Area 11.5 Triple Integrals 11.6 Cylindrical Coordinates 11.7 Spherical Coordinates 11.8 Change of Variables in Multiple Integrals 12 Vector Calculus 12.1 Vector Fields 12.2 Curl and Divergence 12.3 Line Integrals 12.4 Independence of Path and Conservative Vector Fields 12.5 Green's Theorem 12.6 Surface Integrals 12.7 The Divergence Theorem 12.8 Stokes' Theorem 12.9 Applications of Vector Calculus Appendix A Graphs of Additional Polar Equations Appendix B Formal Definition of Limit Appendix C CompleSmith, Robert T. is the author of 'Calculus: Concepts and Connections' with ISBN 9780072826234 and ISBN 0072826231.